LMFDB - Embedded newform 390.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(1,390)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.11416567883$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +1.00000 q^{20} +2.00000 q^{21} -4.00000 q^{22} +2.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +8.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} -4.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +2.00000 q^{38} +1.00000 q^{39} -1.00000 q^{40} +10.0000 q^{41} -2.00000 q^{42} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -2.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -4.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} +2.00000 q^{57} -8.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +4.00000 q^{66} -8.00000 q^{67} +4.00000 q^{68} -2.00000 q^{69} +2.00000 q^{70} -1.00000 q^{72} -6.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -8.00000 q^{77} -1.00000 q^{78} -8.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -8.00000 q^{87} -4.00000 q^{88} -10.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +2.00000 q^{92} -4.00000 q^{93} -2.00000 q^{95} +1.00000 q^{96} -8.00000 q^{97} +3.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	−1.00000	−0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	1.00000	0.223607
$21$	2.00000	0.436436
$22$	−4.00000	−0.852803
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	−2.00000	−0.377964
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	1.00000	0.182574
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	−4.00000	−0.685994
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	2.00000	0.324443
$39$	1.00000	0.160128
$40$	−1.00000	−0.158114
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	−2.00000	−0.308607
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−2.00000	−0.294884
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	−3.00000	−0.428571
$50$	−1.00000	−0.141421
$51$	−4.00000	−0.560112
$52$	−1.00000	−0.138675
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	1.00000	0.136083
$55$	4.00000	0.539360
$56$	2.00000	0.267261
$57$	2.00000	0.264906
$58$	−8.00000	−1.05045
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	−1.00000	−0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	4.00000	0.492366
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	4.00000	0.485071
$69$	−2.00000	−0.240772
$70$	2.00000	0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	−1.00000	−0.117851
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	−6.00000	−0.697486
$75$	−1.00000	−0.115470
$76$	−2.00000	−0.229416
$77$	−8.00000	−0.911685
$78$	−1.00000	−0.113228
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	4.00000	0.433861
$86$	−4.00000	−0.431331
$87$	−8.00000	−0.857690
$88$	−4.00000	−0.426401
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	2.00000	0.208514
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	1.00000	0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	3.00000	0.303046
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	20.0000	1.99007	0.995037	−	0.0995037i	$-0.0317255\pi$
$101$	20.0000	1.99007	0.995037	+	0.0995037i	$0.0317255\pi$
$102$	4.00000	0.396059
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	1.00000	0.0980581
$105$	2.00000	0.195180
$106$	−6.00000	−0.582772
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	−1.00000	−0.0962250
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	−4.00000	−0.381385
$111$	−6.00000	−0.569495
$112$	−2.00000	−0.188982
$113$	−16.0000	−1.50515	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−16.0000	−1.50515	−0.752577	+	0.658505i	$0.771189\pi$
$114$	−2.00000	−0.187317
$115$	2.00000	0.186501
$116$	8.00000	0.742781
$117$	−1.00000	−0.0924500
$118$	12.0000	1.10469
$119$	−8.00000	−0.733359
$120$	1.00000	0.0912871
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	−10.0000	−0.901670
$124$	4.00000	0.359211
$125$	1.00000	0.0894427
$126$	2.00000	0.178174
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	1.00000	0.0877058
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	−4.00000	−0.348155
$133$	4.00000	0.346844
$134$	8.00000	0.691095
$135$	−1.00000	−0.0860663
$136$	−4.00000	−0.342997
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	2.00000	0.170251
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	0	0
$143$	−4.00000	−0.334497
$144$	1.00000	0.0833333
$145$	8.00000	0.664364
$146$	0	0
$147$	3.00000	0.247436
$148$	6.00000	0.493197
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	1.00000	0.0816497
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.00000	0.162221
$153$	4.00000	0.323381
$154$	8.00000	0.644658
$155$	4.00000	0.321288
$156$	1.00000	0.0800641
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	−4.00000	−0.315244
$162$	−1.00000	−0.0785674
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	10.0000	0.780869
$165$	−4.00000	−0.311400
$166$	12.0000	0.931381
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−2.00000	−0.154303
$169$	1.00000	0.0769231
$170$	−4.00000	−0.306786
$171$	−2.00000	−0.152944
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	8.00000	0.606478
$175$	−2.00000	−0.151186
$176$	4.00000	0.301511
$177$	12.0000	0.901975
$178$	10.0000	0.749532
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	1.00000	0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	−2.00000	−0.148250
$183$	2.00000	0.147844
$184$	−2.00000	−0.147442
$185$	6.00000	0.441129
$186$	4.00000	0.293294
$187$	16.0000	1.17004
$188$	0	0
$189$	2.00000	0.145479
$190$	2.00000	0.145095
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−1.00000	−0.0721688
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	8.00000	0.574367
$195$	1.00000	0.0716115
$196$	−3.00000	−0.214286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	−4.00000	−0.284268
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	−1.00000	−0.0707107
$201$	8.00000	0.564276
$202$	−20.0000	−1.40720
$203$	−16.0000	−1.12298
$204$	−4.00000	−0.280056
$205$	10.0000	0.698430
$206$	−4.00000	−0.278693
$207$	2.00000	0.139010
$208$	−1.00000	−0.0693375
$209$	−8.00000	−0.553372
$210$	−2.00000	−0.138013
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	−4.00000	−0.273434
$215$	4.00000	0.272798
$216$	1.00000	0.0680414
$217$	−8.00000	−0.543075
$218$	−4.00000	−0.270914
$219$	0	0
$220$	4.00000	0.269680
$221$	−4.00000	−0.269069
$222$	6.00000	0.402694
$223$	−22.0000	−1.47323	−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000	−1.47323	−0.736614	+	0.676313i	$0.763577\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	16.0000	1.06430
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	2.00000	0.132453
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	−2.00000	−0.131876
$231$	8.00000	0.526361
$232$	−8.00000	−0.525226
$233$	28.0000	1.83434	0.917170	−	0.398495i	$-0.130467\pi$
$233$	28.0000	1.83434	0.917170	+	0.398495i	$0.130467\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	−12.0000	−0.781133
$237$	8.00000	0.519656
$238$	8.00000	0.518563
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	−1.00000	−0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−5.00000	−0.321412
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	−3.00000	−0.191663
$246$	10.0000	0.637577
$247$	2.00000	0.127257
$248$	−4.00000	−0.254000
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	10.0000	0.631194	0.315597	−	0.948893i	$-0.397795\pi$
$251$	10.0000	0.631194	0.315597	+	0.948893i	$0.397795\pi$
$252$	−2.00000	−0.125988
$253$	8.00000	0.502956
$254$	−20.0000	−1.25491
$255$	−4.00000	−0.250490
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	4.00000	0.249029
$259$	−12.0000	−0.745644
$260$	−1.00000	−0.0620174
$261$	8.00000	0.495188
$262$	−14.0000	−0.864923
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	4.00000	0.246183
$265$	6.00000	0.368577
$266$	−4.00000	−0.245256
$267$	10.0000	0.611990
$268$	−8.00000	−0.488678
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	1.00000	0.0608581
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	4.00000	0.242536
$273$	−2.00000	−0.121046
$274$	2.00000	0.120824
$275$	4.00000	0.241209
$276$	−2.00000	−0.120386
$277$	22.0000	1.32185	0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000	1.32185	0.660926	+	0.750451i	$0.270164\pi$
$278$	16.0000	0.959616
$279$	4.00000	0.239474
$280$	2.00000	0.119523
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	0	0
$285$	2.00000	0.118470
$286$	4.00000	0.236525
$287$	−20.0000	−1.18056
$288$	−1.00000	−0.0589256
$289$	−1.00000	−0.0588235
$290$	−8.00000	−0.469776
$291$	8.00000	0.468968
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	−3.00000	−0.174964
$295$	−12.0000	−0.698667
$296$	−6.00000	−0.348743
$297$	−4.00000	−0.232104
$298$	22.0000	1.27443
$299$	−2.00000	−0.115663
$300$	−1.00000	−0.0577350
$301$	−8.00000	−0.461112
$302$	4.00000	0.230174
$303$	−20.0000	−1.14897
$304$	−2.00000	−0.114708
$305$	−2.00000	−0.114520
$306$	−4.00000	−0.228665
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−8.00000	−0.455842
$309$	−4.00000	−0.227552
$310$	−4.00000	−0.227185
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−1.00000	−0.0566139
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	10.0000	0.564333
$315$	−2.00000	−0.112687
$316$	−8.00000	−0.450035
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	6.00000	0.336463
$319$	32.0000	1.79166
$320$	1.00000	0.0559017
$321$	−4.00000	−0.223258
$322$	4.00000	0.222911
$323$	−8.00000	−0.445132
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	−12.0000	−0.664619
$327$	−4.00000	−0.221201
$328$	−10.0000	−0.552158
$329$	0	0
$330$	4.00000	0.220193
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	−12.0000	−0.658586
$333$	6.00000	0.328798
$334$	12.0000	0.656611
$335$	−8.00000	−0.437087
$336$	2.00000	0.109109
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	−1.00000	−0.0543928
$339$	16.0000	0.869001
$340$	4.00000	0.216930
$341$	16.0000	0.866449
$342$	2.00000	0.108148
$343$	20.0000	1.07990
$344$	−4.00000	−0.215666
$345$	−2.00000	−0.107676
$346$	−14.0000	−0.752645
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	−8.00000	−0.428845
$349$	−28.0000	−1.49881	−0.749403	−	0.662114i	$-0.769659\pi$
$349$	−28.0000	−1.49881	−0.749403	+	0.662114i	$0.769659\pi$
$350$	2.00000	0.106904
$351$	1.00000	0.0533761
$352$	−4.00000	−0.213201
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−10.0000	−0.529999
$357$	8.00000	0.423405
$358$	−2.00000	−0.105703
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	22.0000	1.15629
$363$	−5.00000	−0.262432
$364$	2.00000	0.104828
$365$	0	0
$366$	−2.00000	−0.104542
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	2.00000	0.104257
$369$	10.0000	0.520579
$370$	−6.00000	−0.311925
$371$	−12.0000	−0.623009
$372$	−4.00000	−0.207390
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−16.0000	−0.827340
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−8.00000	−0.412021
$378$	−2.00000	−0.102869
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	−2.00000	−0.102598
$381$	−20.0000	−1.02463
$382$	8.00000	0.409316
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	1.00000	0.0510310
$385$	−8.00000	−0.407718
$386$	−12.0000	−0.610784
$387$	4.00000	0.203331
$388$	−8.00000	−0.406138
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	−1.00000	−0.0506370
$391$	8.00000	0.404577
$392$	3.00000	0.151523
$393$	−14.0000	−0.706207
$394$	−10.0000	−0.503793
$395$	−8.00000	−0.402524
$396$	4.00000	0.201008
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−24.0000	−1.20301
$399$	−4.00000	−0.200250
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	−8.00000	−0.399004
$403$	−4.00000	−0.199254
$404$	20.0000	0.995037
$405$	1.00000	0.0496904
$406$	16.0000	0.794067
$407$	24.0000	1.18964
$408$	4.00000	0.198030
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	−10.0000	−0.493865
$411$	2.00000	0.0986527
$412$	4.00000	0.197066
$413$	24.0000	1.18096
$414$	−2.00000	−0.0982946
$415$	−12.0000	−0.589057
$416$	1.00000	0.0490290
$417$	16.0000	0.783523
$418$	8.00000	0.391293
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	2.00000	0.0975900
$421$	−16.0000	−0.779792	−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000	−0.779792	−0.389896	+	0.920859i	$0.627489\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	4.00000	0.194029
$426$	0	0
$427$	4.00000	0.193574
$428$	4.00000	0.193347
$429$	4.00000	0.193122
$430$	−4.00000	−0.192897
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	−1.00000	−0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	8.00000	0.384012
$435$	−8.00000	−0.383571
$436$	4.00000	0.191565
$437$	−4.00000	−0.191346
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−4.00000	−0.190693
$441$	−3.00000	−0.142857
$442$	4.00000	0.190261
$443$	−16.0000	−0.760183	−0.380091	−	0.924949i	$-0.624107\pi$
$443$	−16.0000	−0.760183	−0.380091	+	0.924949i	$0.624107\pi$
$444$	−6.00000	−0.284747
$445$	−10.0000	−0.474045
$446$	22.0000	1.04173
$447$	22.0000	1.04056
$448$	−2.00000	−0.0944911
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−1.00000	−0.0471405
$451$	40.0000	1.88353
$452$	−16.0000	−0.752577
$453$	4.00000	0.187936
$454$	−12.0000	−0.563188
$455$	2.00000	0.0937614
$456$	−2.00000	−0.0936586
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	−28.0000	−1.30835
$459$	−4.00000	−0.186704
$460$	2.00000	0.0932505
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−8.00000	−0.372194
$463$	2.00000	0.0929479	0.0464739	−	0.998920i	$-0.485202\pi$
$463$	2.00000	0.0929479	0.0464739	+	0.998920i	$0.485202\pi$
$464$	8.00000	0.371391
$465$	−4.00000	−0.185496
$466$	−28.0000	−1.29707
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−1.00000	−0.0462250
$469$	16.0000	0.738811
$470$	0	0
$471$	10.0000	0.460776
$472$	12.0000	0.552345
$473$	16.0000	0.735681
$474$	−8.00000	−0.367452
$475$	−2.00000	−0.0917663
$476$	−8.00000	−0.366679
$477$	6.00000	0.274721
$478$	8.00000	0.365911
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	1.00000	0.0456435
$481$	−6.00000	−0.273576
$482$	22.0000	1.00207
$483$	4.00000	0.182006
$484$	5.00000	0.227273
$485$	−8.00000	−0.363261
$486$	1.00000	0.0453609
$487$	38.0000	1.72194	0.860972	−	0.508652i	$-0.169856\pi$
$487$	38.0000	1.72194	0.860972	+	0.508652i	$0.169856\pi$
$488$	2.00000	0.0905357
$489$	−12.0000	−0.542659
$490$	3.00000	0.135526
$491$	38.0000	1.71492	0.857458	−	0.514554i	$-0.172042\pi$
$491$	38.0000	1.71492	0.857458	+	0.514554i	$0.172042\pi$
$492$	−10.0000	−0.450835
$493$	32.0000	1.44121
$494$	−2.00000	−0.0899843
$495$	4.00000	0.179787
$496$	4.00000	0.179605
$497$	0	0
$498$	−12.0000	−0.537733
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	1.00000	0.0447214
$501$	12.0000	0.536120
$502$	−10.0000	−0.446322
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	2.00000	0.0890871
$505$	20.0000	0.889988
$506$	−8.00000	−0.355643
$507$	−1.00000	−0.0444116
$508$	20.0000	0.887357
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	4.00000	0.177123
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	2.00000	0.0883022
$514$	8.00000	0.352865
$515$	4.00000	0.176261
$516$	−4.00000	−0.176090
$517$	0	0
$518$	12.0000	0.527250
$519$	−14.0000	−0.614532
$520$	1.00000	0.0438529
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−8.00000	−0.350150
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	14.0000	0.611593
$525$	2.00000	0.0872872
$526$	18.0000	0.784837
$527$	16.0000	0.696971
$528$	−4.00000	−0.174078
$529$	−19.0000	−0.826087
$530$	−6.00000	−0.260623
$531$	−12.0000	−0.520756
$532$	4.00000	0.173422
$533$	−10.0000	−0.433148
$534$	−10.0000	−0.432742
$535$	4.00000	0.172935
$536$	8.00000	0.345547
$537$	−2.00000	−0.0863064
$538$	4.00000	0.172452
$539$	−12.0000	−0.516877
$540$	−1.00000	−0.0430331
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	28.0000	1.20270
$543$	22.0000	0.944110
$544$	−4.00000	−0.171499
$545$	4.00000	0.171341
$546$	2.00000	0.0855921
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−2.00000	−0.0854358
$549$	−2.00000	−0.0853579
$550$	−4.00000	−0.170561
$551$	−16.0000	−0.681623
$552$	2.00000	0.0851257
$553$	16.0000	0.680389
$554$	−22.0000	−0.934690
$555$	−6.00000	−0.254686
$556$	−16.0000	−0.678551
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	−4.00000	−0.169334
$559$	−4.00000	−0.169182
$560$	−2.00000	−0.0845154
$561$	−16.0000	−0.675521
$562$	10.0000	0.421825
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	−16.0000	−0.673125
$566$	−28.0000	−1.17693
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	−2.00000	−0.0837708
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	−4.00000	−0.167248
$573$	8.00000	0.334205
$574$	20.0000	0.834784
$575$	2.00000	0.0834058
$576$	1.00000	0.0416667
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	1.00000	0.0415945
$579$	−12.0000	−0.498703
$580$	8.00000	0.332182
$581$	24.0000	0.995688
$582$	−8.00000	−0.331611
$583$	24.0000	0.993978
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	6.00000	0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	3.00000	0.123718
$589$	−8.00000	−0.329634
$590$	12.0000	0.494032
$591$	−10.0000	−0.411345
$592$	6.00000	0.246598
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	4.00000	0.164122
$595$	−8.00000	−0.327968
$596$	−22.0000	−0.901155
$597$	−24.0000	−0.982255
$598$	2.00000	0.0817861
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	8.00000	0.326056
$603$	−8.00000	−0.325785
$604$	−4.00000	−0.162758
$605$	5.00000	0.203279
$606$	20.0000	0.812444
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	2.00000	0.0811107
$609$	16.0000	0.648353
$610$	2.00000	0.0809776
$611$	0	0
$612$	4.00000	0.161690
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	0	0
$615$	−10.0000	−0.403239
$616$	8.00000	0.322329
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	4.00000	0.160904
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	4.00000	0.160644
$621$	−2.00000	−0.0802572
$622$	16.0000	0.641542
$623$	20.0000	0.801283
$624$	1.00000	0.0400320
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	8.00000	0.319489
$628$	−10.0000	−0.399043
$629$	24.0000	0.956943
$630$	2.00000	0.0796819
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	8.00000	0.318223
$633$	4.00000	0.158986
$634$	2.00000	0.0794301
$635$	20.0000	0.793676
$636$	−6.00000	−0.237915
$637$	3.00000	0.118864
$638$	−32.0000	−1.26689
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000	0.157867
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	−4.00000	−0.157622
$645$	−4.00000	−0.157500
$646$	8.00000	0.314756
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	−1.00000	−0.0392837
$649$	−48.0000	−1.88416
$650$	1.00000	0.0392232
$651$	8.00000	0.313545
$652$	12.0000	0.469956
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	4.00000	0.156412
$655$	14.0000	0.547025
$656$	10.0000	0.390434
$657$	0	0
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	−4.00000	−0.155700
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	10.0000	0.388661
$663$	4.00000	0.155347
$664$	12.0000	0.465690
$665$	4.00000	0.155113
$666$	−6.00000	−0.232495
$667$	16.0000	0.619522
$668$	−12.0000	−0.464294
$669$	22.0000	0.850569
$670$	8.00000	0.309067
$671$	−8.00000	−0.308837
$672$	−2.00000	−0.0771517
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	26.0000	1.00148
$675$	−1.00000	−0.0384900
$676$	1.00000	0.0384615
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	−16.0000	−0.614476
$679$	16.0000	0.614024
$680$	−4.00000	−0.153393
$681$	−12.0000	−0.459841
$682$	−16.0000	−0.612672
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	−2.00000	−0.0764719
$685$	−2.00000	−0.0764161
$686$	−20.0000	−0.763604
$687$	−28.0000	−1.06827
$688$	4.00000	0.152499
$689$	−6.00000	−0.228582
$690$	2.00000	0.0761387
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	14.0000	0.532200
$693$	−8.00000	−0.303895
$694$	−16.0000	−0.607352
$695$	−16.0000	−0.606915
$696$	8.00000	0.303239
$697$	40.0000	1.51511
$698$	28.0000	1.05982
$699$	−28.0000	−1.05906
$700$	−2.00000	−0.0755929
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	−1.00000	−0.0377426
$703$	−12.0000	−0.452589
$704$	4.00000	0.150756
$705$	0	0
$706$	−6.00000	−0.225813
$707$	−40.0000	−1.50435
$708$	12.0000	0.450988
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	10.0000	0.374766
$713$	8.00000	0.299602
$714$	−8.00000	−0.299392
$715$	−4.00000	−0.149592
$716$	2.00000	0.0747435
$717$	8.00000	0.298765
$718$	−24.0000	−0.895672
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	1.00000	0.0372678
$721$	−8.00000	−0.297936
$722$	15.0000	0.558242
$723$	22.0000	0.818189
$724$	−22.0000	−0.817624
$725$	8.00000	0.297113
$726$	5.00000	0.185567
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−2.00000	−0.0741249
$729$	1.00000	0.0370370
$730$	0	0
$731$	16.0000	0.591781
$732$	2.00000	0.0739221
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	4.00000	0.147643
$735$	3.00000	0.110657
$736$	−2.00000	−0.0737210
$737$	−32.0000	−1.17874
$738$	−10.0000	−0.368105
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	6.00000	0.220564
$741$	−2.00000	−0.0734718
$742$	12.0000	0.440534
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	4.00000	0.146647
$745$	−22.0000	−0.806018
$746$	−14.0000	−0.512576
$747$	−12.0000	−0.439057
$748$	16.0000	0.585018
$749$	−8.00000	−0.292314
$750$	1.00000	0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	0	0
$753$	−10.0000	−0.364420
$754$	8.00000	0.291343
$755$	−4.00000	−0.145575
$756$	2.00000	0.0727393
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	14.0000	0.508503
$759$	−8.00000	−0.290382
$760$	2.00000	0.0725476
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	20.0000	0.724524
$763$	−8.00000	−0.289619
$764$	−8.00000	−0.289430
$765$	4.00000	0.144620
$766$	20.0000	0.722629
$767$	12.0000	0.433295
$768$	−1.00000	−0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	8.00000	0.288300
$771$	8.00000	0.288113
$772$	12.0000	0.431889
$773$	26.0000	0.935155	0.467578	−	0.883952i	$-0.345127\pi$
$773$	26.0000	0.935155	0.467578	+	0.883952i	$0.345127\pi$
$774$	−4.00000	−0.143777
$775$	4.00000	0.143684
$776$	8.00000	0.287183
$777$	12.0000	0.430498
$778$	−20.0000	−0.717035
$779$	−20.0000	−0.716574
$780$	1.00000	0.0358057
$781$	0	0
$782$	−8.00000	−0.286079
$783$	−8.00000	−0.285897
$784$	−3.00000	−0.107143
$785$	−10.0000	−0.356915
$786$	14.0000	0.499363
$787$	−28.0000	−0.998092	−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000	−0.998092	−0.499046	+	0.866575i	$0.666316\pi$
$788$	10.0000	0.356235
$789$	18.0000	0.640817
$790$	8.00000	0.284627
$791$	32.0000	1.13779
$792$	−4.00000	−0.142134
$793$	2.00000	0.0710221
$794$	−6.00000	−0.212932
$795$	−6.00000	−0.212798
$796$	24.0000	0.850657
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−10.0000	−0.353333
$802$	30.0000	1.05934
$803$	0	0
$804$	8.00000	0.282138
$805$	−4.00000	−0.140981
$806$	4.00000	0.140894
$807$	4.00000	0.140807
$808$	−20.0000	−0.703598
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−1.00000	−0.0351364
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	−16.0000	−0.561490
$813$	28.0000	0.982003
$814$	−24.0000	−0.841200
$815$	12.0000	0.420342
$816$	−4.00000	−0.140028
$817$	−8.00000	−0.279885
$818$	18.0000	0.629355
$819$	2.00000	0.0698857
$820$	10.0000	0.349215
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−2.00000	−0.0697580
$823$	−20.0000	−0.697156	−0.348578	−	0.937280i	$-0.613335\pi$
$823$	−20.0000	−0.697156	−0.348578	+	0.937280i	$0.613335\pi$
$824$	−4.00000	−0.139347
$825$	−4.00000	−0.139262
$826$	−24.0000	−0.835067
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	2.00000	0.0695048
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	12.0000	0.416526
$831$	−22.0000	−0.763172
$832$	−1.00000	−0.0346688
$833$	−12.0000	−0.415775
$834$	−16.0000	−0.554035
$835$	−12.0000	−0.415277
$836$	−8.00000	−0.276686
$837$	−4.00000	−0.138260
$838$	−14.0000	−0.483622
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	−2.00000	−0.0690066
$841$	35.0000	1.20690
$842$	16.0000	0.551396
$843$	10.0000	0.344418
$844$	−4.00000	−0.137686
$845$	1.00000	0.0344010
$846$	0	0
$847$	−10.0000	−0.343604
$848$	6.00000	0.206041
$849$	−28.0000	−0.960958
$850$	−4.00000	−0.137199
$851$	12.0000	0.411355
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	−4.00000	−0.136877
$855$	−2.00000	−0.0683986
$856$	−4.00000	−0.136717
$857$	−16.0000	−0.546550	−0.273275	−	0.961936i	$-0.588107\pi$
$857$	−16.0000	−0.546550	−0.273275	+	0.961936i	$0.588107\pi$
$858$	−4.00000	−0.136558
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	4.00000	0.136399
$861$	20.0000	0.681598
$862$	−16.0000	−0.544962
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	1.00000	0.0340207
$865$	14.0000	0.476014
$866$	26.0000	0.883516
$867$	1.00000	0.0339618
$868$	−8.00000	−0.271538
$869$	−32.0000	−1.08553
$870$	8.00000	0.271225
$871$	8.00000	0.271070
$872$	−4.00000	−0.135457
$873$	−8.00000	−0.270759
$874$	4.00000	0.135302
$875$	−2.00000	−0.0676123
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	−8.00000	−0.269987
$879$	6.00000	0.202375
$880$	4.00000	0.134840
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	3.00000	0.101015
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−4.00000	−0.134535
$885$	12.0000	0.403376
$886$	16.0000	0.537531
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	6.00000	0.201347
$889$	−40.0000	−1.34156
$890$	10.0000	0.335201
$891$	4.00000	0.134005
$892$	−22.0000	−0.736614
$893$	0	0
$894$	−22.0000	−0.735790
$895$	2.00000	0.0668526
$896$	2.00000	0.0668153
$897$	2.00000	0.0667781
$898$	18.0000	0.600668
$899$	32.0000	1.06726
$900$	1.00000	0.0333333
$901$	24.0000	0.799556
$902$	−40.0000	−1.33185
$903$	8.00000	0.266223
$904$	16.0000	0.532152
$905$	−22.0000	−0.731305
$906$	−4.00000	−0.132891
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	12.0000	0.398234
$909$	20.0000	0.663358
$910$	−2.00000	−0.0662994
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	2.00000	0.0662266
$913$	−48.0000	−1.58857
$914$	16.0000	0.529233
$915$	2.00000	0.0661180
$916$	28.0000	0.925146
$917$	−28.0000	−0.924641
$918$	4.00000	0.132020
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	−2.00000	−0.0659380
$921$	0	0
$922$	−30.0000	−0.987997
$923$	0	0
$924$	8.00000	0.263181
$925$	6.00000	0.197279
$926$	−2.00000	−0.0657241
$927$	4.00000	0.131377
$928$	−8.00000	−0.262613
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	4.00000	0.131165
$931$	6.00000	0.196642
$932$	28.0000	0.917170
$933$	16.0000	0.523816
$934$	−12.0000	−0.392652
$935$	16.0000	0.523256
$936$	1.00000	0.0326860
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	−16.0000	−0.522419
$939$	10.0000	0.326338
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−10.0000	−0.325818
$943$	20.0000	0.651290
$944$	−12.0000	−0.390567
$945$	2.00000	0.0650600
$946$	−16.0000	−0.520205
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	2.00000	0.0648886
$951$	2.00000	0.0648544
$952$	8.00000	0.259281
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	−6.00000	−0.194257
$955$	−8.00000	−0.258874
$956$	−8.00000	−0.258738
$957$	−32.0000	−1.03441
$958$	−24.0000	−0.775405
$959$	4.00000	0.129167
$960$	−1.00000	−0.0322749
$961$	−15.0000	−0.483871
$962$	6.00000	0.193448
$963$	4.00000	0.128898
$964$	−22.0000	−0.708572
$965$	12.0000	0.386294
$966$	−4.00000	−0.128698
$967$	14.0000	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$967$	14.0000	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$968$	−5.00000	−0.160706
$969$	8.00000	0.256997
$970$	8.00000	0.256865
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	−1.00000	−0.0320750
$973$	32.0000	1.02587
$974$	−38.0000	−1.21760
$975$	1.00000	0.0320256
$976$	−2.00000	−0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	12.0000	0.383718
$979$	−40.0000	−1.27841
$980$	−3.00000	−0.0958315
$981$	4.00000	0.127710
$982$	−38.0000	−1.21263
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	10.0000	0.318788
$985$	10.0000	0.318626
$986$	−32.0000	−1.01909
$987$	0	0
$988$	2.00000	0.0636285
$989$	8.00000	0.254385
$990$	−4.00000	−0.127128
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	−4.00000	−0.127000
$993$	10.0000	0.317340
$994$	0	0
$995$	24.0000	0.760851
$996$	12.0000	0.380235
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	14.0000	0.443162
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.2.a.b.1.1	✓	1
3.2	odd	2		1170.2.a.j.1.1		1
4.3	odd	2		3120.2.a.y.1.1		1
5.2	odd	4		1950.2.e.m.1249.1		2
5.3	odd	4		1950.2.e.m.1249.2		2
5.4	even	2		1950.2.a.ba.1.1		1
12.11	even	2		9360.2.a.v.1.1		1
13.5	odd	4		5070.2.b.f.1351.2		2
13.8	odd	4		5070.2.b.f.1351.1		2
13.12	even	2		5070.2.a.n.1.1		1
15.2	even	4		5850.2.e.h.5149.2		2
15.8	even	4		5850.2.e.h.5149.1		2
15.14	odd	2		5850.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.b.1.1	✓	1	1.1	even	1	trivial
1170.2.a.j.1.1		1	3.2	odd	2
1950.2.a.ba.1.1		1	5.4	even	2
1950.2.e.m.1249.1		2	5.2	odd	4
1950.2.e.m.1249.2		2	5.3	odd	4
3120.2.a.y.1.1		1	4.3	odd	2
5070.2.a.n.1.1		1	13.12	even	2
5070.2.b.f.1351.1		2	13.8	odd	4
5070.2.b.f.1351.2		2	13.5	odd	4
5850.2.a.s.1.1		1	15.14	odd	2
5850.2.e.h.5149.1		2	15.8	even	4
5850.2.e.h.5149.2		2	15.2	even	4
9360.2.a.v.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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